
theorem
  for a,x be Integer, p be Prime st
  a,p are_coprime & a,x*x are_congruent_mod p holds x,p are_coprime
proof
  let a,x be Integer;
  let p be Prime;
  assume that
A1: a,p are_coprime and
A2: a,x*x are_congruent_mod p;
  x*x,p are_coprime by A1,A2,Th31;
  then
A3: x*x gcd p = 1 by INT_2:def 3;
  assume
A4: not x,p are_coprime;
A5: (x gcd p) divides p by INT_2:21;
A6: x gcd p >= 0;
  (x gcd p) divides (x*x) by INT_2:2,21;
  then (x gcd p) = 1 or (x gcd p) = -1 by A3,A5,INT_2:13,22;
  hence contradiction by A4,A6,INT_2:def 3;
end;
